In a recent classroom project, we studied the growth of a fruit fly population in a favorable environment. In particular, we assumed that there was no limit, in terms of the resources available, to the population size. But in a real-world situation, of course, there will be limited amounts of food, water and space, and the population will not be able to grow without bound. In this lab, we examine the problem of modeling the population as a function of time in a setting where there is a maximum population that the environment will support.
In contrast to the classroom project, instead of starting from data, we will start from an assumption about the nature of the growth rate. In further contrast to the project, where we used a fixed time step of one day, we will consider what happens as our time step gets smaller and smaller. Our purpose is to discover a description of the population function that follows from our assumption about the growth rate and from the observation that "rate equals rise divided by run."
Read this carefully! Talk to your lab partner about it. Scribble on a sheet of paper. Ask questions. Do whatever you need to do to understand this part.
You will type your answers into this Mathcad document as you go along.
***Save your work often!! ***
1. When there is a maximum population M that the environment will support, biologists often assume that the rate of change of the population is proportional to the product of the population and [M minus the population]. (In your science books, you may see "proportional to the product of" expressed as "jointly proportional to.") In other words,
rate of change = c * pop * (M - pop)
where c is the constant of proportionality.
The reason for the assumption is this: When the population is small, [M minus the population] is essentially the same thing as M, and then the proportionality is to the population itself, the "natural growth" situation considered in the class project:
rate of change @ (cM) * pop when pop is small.
However, when the population gets close to M, the factor [M - pop] is close to zero, so the growth rate is close to zero, and further growth is shut down. Thus, this factor expresses in a natural way the effect of the limited environment.
Let's try to convert this assumption about the growth rate into a model for the population as a function of time. As in the classroom project, we'll assume that the initial population is 111 fruit flies, and we will add a new assumption that the maximum population supported by the laboratory environment is 1000. We also need a proportionality constant c; for reasons to be investigated later, we will take c = 9.8 x 10-5. Our objective is to describe the population over the next 100 days.
For our first model, we'll describe the population at 20 equally spaced points in time; i.e., at intervals of 5 days. Then
We label the population values pop0, pop1, pop2, and so on. We can label all the population values at once by using "popk" as k varies between 0 and 20. The corresponding times are labeled by "timek."
How can we calculate pop1, the population at time1 = 5 days? We know that the rate of change of the population is Dpop/Dt = rise/run, where the rise is pop1 - pop0, and the run is 5 days. Recall that the rate of change is the same as the slope. If we denote rise/run here by slope0, then our assumption on the rate of change of this population leads to
rate of change = slope0 = c * pop0 * (M-pop0).
We also know that
or slope0 = (pop1 - pop0)/run.
Solve this equation for pop1:
pop1 =
Ask your instructor to check the formula for pop1 before you go on.
Now that we know pop1, we can use it to find slope1 and pop2, by going through
the same steps you just went through:
slope1 = c*pop1*(M-pop1)
pop2 = pop1 + run * slope1
And so on. In general, we have :
So in this way we can calculate the population at the n times t = 5 days, t = 10 days, etc., up to t = 100 days.
Study the plots of slope and population as functions of time in the graph boxes on the following page. Remember that the computer has values for pop0, pop1, pop2, etc. stored, and it can find the slope at any of these values.
3. Go back to the beginning of the worksheet, and change the number of time steps from 20 to 40. How often are we computing the population now? Again find values for the population at 20 and 100 days and record your answers in text below; plot the slopes and populations.
Change the number of time steps from 40 to 80. At this point you will probably want to change the graph style from points to lines, and get rid of those X's. Find values for the population at 20 and 100 days, and plot the slopes and populations. Record your findings.
Change the number of time steps from 80 to 160. Once more, find values for the population at 20 and 100 days, and plot the slopes and populations.
What happens to the models as we take more and more time steps in the same 100-day interval? Which model do you think most accurately portrays the real situation? Why?
4. Estimate the time at which the fly population is increasing most rapidly, and estimate the size of the fly population at this time. Explain your reasoning.
